Time Complexity

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Time Complexity Slides based on RPI CSCI
2400 Thanks to Costas Busch
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Question so far
Computable? Yes or no?
This lecture Computable in reasonable
time? Goal Low time complexity
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Consider a decidable (recursive) language
which is decided by a Turing Machine
For any string the computation
of terminates (in an accept of reject state) in a
finite amount of transitions
Accept or Reject
Time transitions followed
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Max time to accept a string of length
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Language class
Languages for which there is a deterministic
Turing Machine that decides each string of the
language within time
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Language classes for 1) an 2) anbn 3)
membership problem for context free languages ?
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Examples
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Big-oh notation
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Example The membership problem
for context free languages
(CYK - algorithm)
Polynomial time
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Theorem
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Polynomial time algorithms
For every constant
Represent tractable algorithms
For small we can compute the result fast
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Example word w, !w!10
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The class
for all

• Polynomial time

• All tractable problems

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CYK-algorithm
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Exponential time algorithms
Represent intractable algorithms
Some problem instances may take centuries to
solve Making computers faster doesn't help. Why?
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Example
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Example the Hamiltonian Path Problem
s
t
Question is there a Hamiltonian path
from s to t?
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s
t
YES!
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Algorithm?
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A solution search exhaustively all paths
L ltG,s,tgt there is a Hamiltonian path
in G from s to t
Exponential time
Intractable problem
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The clique problem
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The clique problem Exponential
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Example The Satisfiability Problem
Boolean expressions in Conjunctive Normal Form
Variables
Question is the expression satisfiable?
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(not x1 or x2) and (x1 or x3) (x1 or x2) and
(not x1) and (not x2)
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Example
Satisfiable
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Example
Not satisfiable
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exponential
Algorithm search exhaustively all the
possible binary values of the variables
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Non-Determinism
Language class
A Non-Deterministic Turing Machine accepts each
string of length in time
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Example
Non-Deterministic Algorithm to accept a string

• Use a two-tape Turing machine
• Guess the middle of the string
• and copy on the second tape
• Compare the two tapes

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Time needed

• Use a two-tape Turing machine
• Guess the middle of the string
• and copy on the second tape
• Compare the two tapes

Total time
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(No Transcript)
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In a similar way we define the class
for any time function
Examples
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Non-Deterministic Polynomial time algorithms
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The class
for all
Non-Deterministic Polynomial time
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The satisfiability problem
Example
Non-Deterministic algorithm

• Guess an assignment of the variables

• Check if this is a satisfying assignment

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Time for variables (or length of
expression?)

• Guess an assignment of the variables

• Check if this is a satisfying assignment

Total time
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The satisfiability problem is an - Problem
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Observation
Deterministic Polynomial
Non-Deterministic Polynomial
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Open Problem
WE DO NOT KNOW THE ANSWER
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Open Problem
Example Does the Satisfiability problem
have a polynomial time
deterministic algorithm?
WE DO NOT KNOW THE ANSWER